Neumann eigenvalues are known to lack monotonicity with respect to domain inclusion. As a result, it makes sense to study the two shape optimization problems $\min\{\mu_k(\Omega) \;:\; \Omega\text{ convex},\; \Omega\subset D\}$ (for a given box $D$) and $\max\{\mu_k(\Omega) \;:\; \Omega\text{ convex},\; \omega \subset\Omega\}$ (for a given obstacle $\omega$). In this talk, we address the problem of the existence of a solution to these two problems in two dimensions, and we give some qualitative properties of the solutions. We also introduce the notion of self-domains, which are domains solutions of these extremal problems for themselves, and give examples of the disk and the square. Numerical simulations are also presented.